solve in the simplest exact form the inequality (lnx)^2-6lnx+8 <= 0

Let's solve the inequality \((\ln x)^2 - 6 \ln x + 8 \leq 0\).

1. **Substitute \( \ln x = t \):**

   Let \( t = \ln x \). The inequality becomes:
   \[
   t^2 - 6t + 8 \leq 0
   \]

2. **Solve the quadratic inequality:**

   Factor the quadratic expression:
   \[
   t^2 - 6t + 8 = (t - 2)(t - 4)
   \]
   So the inequality is:
   \[
   (t - 2)(t - 4) \leq 0
   \]

3. **Determine the solution intervals for \( t \):**

   The roots of the equation \( (t - 2)(t - 4) = 0 \) are \( t = 2 \) and \( t = 4 \). To determine where the product is non-positive, consider the sign of \( (t - 2)(t - 4) \) in each of the following intervals:

   - For \( t < 2 \): Both \( t - 2 \) and \( t - 4 \) are negative, so \( (t - 2)(t - 4) > 0 \).
   - For \( 2 \leq t \leq 4 \): \( t - 2 \geq 0 \) and \( t - 4 \leq 0 \), so \( (t - 2)(t - 4) \leq 0 \).
   - For \( t > 4 \): Both \( t - 2 \) and \( t - 4 \) are positive, so \( (t - 2)(t - 4) > 0 \).

   Therefore, the solution for \( t \) is:
   \[
   2 \leq t \leq 4
   \]

4. **Back-substitute \( t = \ln x \):**

   Since \( t = \ln x \), the inequality becomes:
   \[
   2 \leq \ln x \leq 4
   \]

   Exponentiate both sides to solve for \( x \):
   \[
   e^2 \leq x \leq e^4
   \]

5. **Conclusion:**

   The solution in terms of \( x \) is:
   \[
   e^2 \leq x \leq e^4
   \]

This is the simplest exact form of the solution. Do you want any details or have any questions?

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1. What is the meaning of the natural logarithm function?
2. How can we graphically represent the solution of the inequality?
3. What are the properties of quadratic inequalities?
4. How do we solve a quadratic equation using factorization?
5. What are some other methods for solving quadratic inequalities?

**Tip:** When dealing with logarithmic inequalities, always check the domain of the logarithmic function, as it is defined only for positive arguments.

solve in the simplest exact form the inequality (lnx)^2 - 6lnx + 8 <= 0

This detailed guide walks through solving the inequality (lnx)^2 - 6lnx + 8 <= 0, covering key concepts like logarithms, quadratic inequalities, and exponentiation. Suitable for high school students in Grades 10-12.

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